       To achieve the desired fifth-order accuracy when the
first and second derivatives vanish but the third derivative is non-zero,
a modified WENO-P scheme called MWENO-P is proposed here. The Taylor's
expansion of the operators \(L_{n,k}f\)  defined in (REF ,
) by using Theorem \(3.2\)  of [1]} are
\(L_{1,0}f & = & \Delta xf_{j+\frac{1}{2}}^{(1)}-\frac{23}{24}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}+\Delta x^{4}f_{j+\frac{1}{2}}^{(4)}+O(\Delta x^{5}),\nonumber \\L_{1,1}f & = & \Delta xf_{j+\frac{1}{2}}^{(1)}+\frac{1}{24}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}+O(\Delta x^{5}),\nonumber \\L_{1,2}f & = & \Delta xf_{j+\frac{1}{2}}^{(1)}+\frac{1}{24}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}+O(\Delta x^{5}),\nonumber \\L_{2,0}f & = & \Delta x^{2}f_{j+\frac{1}{2}}^{(2)}-\frac{3}{2}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}+\frac{29}{384}\Delta x^{4}f_{j+\frac{1}{2}}^{(4)}+O(\Delta x^{5}),\\L_{2,1}f & = & \Delta x^{2}f_{j+\frac{1}{2}}^{(2)}-\frac{1}{2}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}+\frac{5}{24}\Delta x^{4}f_{j+\frac{1}{2}}^{(4)}+O(\Delta x^{5}),\nonumber \\L_{2,2}f & = & \Delta x^{2}f_{j+\frac{1}{2}}^{(2)}+\frac{1}{2}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}+\frac{5}{24}\Delta x^{4}f_{j+\frac{1}{2}}^{(4)}+O(\Delta x^{5}).\nonumber \) 
